6TEKA Category Theory, winter semester 2023/2024
HOMEWORKS
Homework 1
Does there exist a category consisting of:
a) 4 objects and 3 arrows;
b) 3 objects and 4 arrows?
Homework 2
For which natural values of \(n\) the following is true: there are "more" (in
any conceivable sense) categories with \(n+1\) objects than categories with
\(n\) objects?
Homework 3
Denote by \(\mathbf S_X\) the category associated to a set \(X\) (by considering
\(X\) as the set of objects, and identity maps on elements of \(X\) as the set
of arrows), and by \(\mathbf C_M\) the category associated to a monoid \(M\),
as discussed in the class. Describe all the cases when one of these categories
is a subcategory of the other.
Homework 4
Describe all covariant functors from the category \(\mathbf{C}_M\) associated to
a fixed monoid \(M\), to \(\mathbf{Set}\).
Homework 5
Is it possible that a functor between two categories is simultaneously covariant
and contravariant?
Homework 6
Let \(U\) be a fixed vector space over a field \(K\). Define a functor
\(\mathbf{Vect}_K \to \mathbf{Vect}_K\) sending a vector space \(V\) to
\(Hom_K(V,U)\). Prove that this is a functor. Is this functor covariant or
contravariant? How this construction can be generalized?
Homework 7
Enumerate, up to isomorphism, all categories containing exactly 3 arrows, and
for each of them find all subcategories, up to isomorphism.
Homework 8
Recall that a constant functor between two categories \(\mathbb C\) and
\(\mathbb D\) is a functor sending each object of \(\mathbb C\) to a fixed
object \(A\) in \(\mathbb D\), and each arrow in \(\mathbb C\), to \(1_A\).
Describe all situations when a constant functor is isomorphism of categories.
Homework 9
Prove that the forgetful functor from \(\mathbf{Group}\) to \(\mathbf{Set}\) is
not an isomorphism.
Homework 10
Which of the three categories: \(\mathbb C_M\) for a fixed monoid \(M\),
\(\mathbb S_X\) for a fixed set \(X\), and \(\mathbf{Set}\) are equivalent?
Homework 13
Prove that there exists a natural transformation between the identity functor
\(id_{\mathbf{Set}}: \mathbf{Set} \to \mathbf{Set}\), and the power set functor
\(P: \mathbf{Set} \to \mathbf{Set}\) (i.e., the functor assigning to any set
\(X\) its power set \(P(X)\)). Is this natural transformation a natural
equivalence?
Last modified: Mon Jan 15 09:07:18 CET 2024